Generalizations of some results about the regularity properties of an additive representation function

Abstract

Let $A=\{a_1,a_2,\dots \}$ $(a_1<a_2<\dots )$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_x+a_y=n$ $(a_x,a_y\in A)$. P. Erdős, A. Sárközy and V. T. Sós proved that if ${\lim }_{N\to \infty }\frac{B(A,N)}{\sqrt{N}}=+\infty$ then $|{\Delta }_1(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and ${\Delta }_l$ denotes the $l$-th difference. Their result was extended to ${\Delta }_l(R_{A,2}(n))$ for any fixed $l\ge 2$. In this paper we give further generalizations of this problem.